Machinery's Handbook, 31st Edition
Calculus
133
Step 2. The first derivative, f ′ ( x ), of x 2 - 101 is 2 x, as stated previously, so that f ′ (10) = 2(10) = 20. Then, r 2 = r 1 - f ( r 1 ) /f ′ ( r 1 ) = 10 - ( - 1)/20 = 10 + 0.05 = 10.05 Check: 10.05 2 = 101.0025; a calculator determination of 101 gives 10.0498756; the error using Newton’s method is 0.0001244. Step 3. The next, better approximation is r 3 r 2 f r 2 ( ) f ′ r 2 ⁄ ( ) – 10.05 f 10.05 ( ) f ′ 10.05 ( ) ⁄ – = = 10.05 10.05 2 – 101 ( ) 2 10.05 ( ) ⁄ – = = 10.049875 Check: 10.049875 2 100.9999875 error ; 0.0000125 = = Closer approximations result from subsequent applications of the algorithm. Formulas for Differential and Integral Calculus.— The following are formulas for ob- taining the derivatives and integrals of basic mathematical functions. In these formulas, the letters a , b , and c denote constants; the letter x denotes a variable; and the letters u and v denote functions of the variable x . The expression d/dx means the derivative with respect to x , and as such applies to whatever expression in parentheses follows it. Thus, d ( cx ) /dx means the derivative with respect to the variable x of the product cx where c is a constant. Table of Derivatives and Integrals Derivatives Integrals
dx d dx d
cdx ∫ = cx +constant
c ( ) = 0
x ( ) = 1
1 dx ∫ = x + C
n +1
n –1 , for all real numbers n
dx d
x
x n ( ) = nx
n +1 ------ + C , for all real numbers n ≠ –1
=
x n dx ∫
x ( ) = –
d 1
1 x 2 ----- + C
dx ax b + -------- ∫ = 1 a
--ln ax b + + C
dx
dx d x log a (
1 x ln a ----------
1 x -- dx ∫ = ln | x | + C
) =
dx d dx d
e x ( ) =
e x dx ∫ = e
x
x + C
e
a e ax + C
e ax ( ) = ae ax
e ax dx =
a x dx ∫ = a x a ln
a x ( ) = a x a ln
dx d
------ + C
1 2 x -----
dx d
dx
x ( ) =
x --- ∫ = 2 x + C
dx d ln x ( ) = 1 x --
x ln dx ∫ = x ln x – x + C
dx d
x sin + C
x sin ( ) =
x cos dx ∫ =
x cos
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